Hodge theory lecture 16: Currents and the Poincar - - PowerPoint PPT Presentation

Hodge theory, lecture 16

• M. Verbitsky

Hodge theory

lecture 16: Currents and the Poincar´ e-Dolbeault-Grothendieck lemma NRU HSE, Moscow Misha Verbitsky, March 21, 2018

1

Hodge theory, lecture 16

• M. Verbitsky

Generalized functions DEFINITION: Let V be a vector space equipped with a collection of norms (or seminorms) | · |i, i = 0, 1, 2, ... and a topology which is given by the metric d(x, y) =

∞

• i=0

2−i min(|x − y|i, 1), assumed to be non-degenerate. The space V is called a Fr´ echet space if this metric is complete. REMARK: Completeness is equivalent to convergence of any sequence {ai} which is fundamental with respect to all the (semi-)norms | · |i. REMARK: A sequence converges in the Fr´ echet topology given by d ⇔ it converges in any of the (semi-)norms | · |i. DEFINITION: Let M be a Riemannian manifold, and ∇i : C∞(M) − → Λ1(M)⊗i the iterated connection. Topology Ck on the space C∞

c (M) of functions with

compact support is defined by the norm |ϕ|Ck := sup

M k

• i=0

|∇iϕ|. 2

Hodge theory, lecture 16

• M. Verbitsky

Generalized functions (2) DEFINITION: The space of test-functions with compact support is the space of functions with compact support and a metric d(x, y) =

∞

• i=0

2−i min(|x − y|Ci, 1).

• f uniform convergence of all derivatives.

EXERCISE: Prove that the space of test-functions with support in a compact set K ⊂ M is a Fr´ echet space. DEFINITION: Generalized function (also called distribution) is a func- tional on the space of test-function which is continuous in one of the Ci- topologies on the space C∞(M)K of functions with support in any compact K ⊂ M. EXAMPLE: Delta-function δz is a functional mapping ϕ ∈ C∞

c (M) to ϕ(z),

for a given point z ∈ M. Delta-function is continuous in the topology C0, its derivative is continuous in C1 and so on. 3

Hodge theory, lecture 16

• M. Verbitsky

Currents on complex manifolds REMARK: The Ci-topology is defined on the space of sections of any vector bundle B over using the same formula. It depends on the choice of the metric

• n M and on B, but the induced topology is clearly independent from

this choice. DEFINITION: The space of test-forms of type (p, q) on a complex manifold is the space Λp,q

c (M) with compact support, equipped with the

Fr´ echet topology as on the test-functions. DEFINITION: A (p, q)-current on a complex n-dimensional manifold is a functional θ on the space Λn−p,n−q

c

(M) of forms with compact support, such that for any compact set K ⊂ M there exists i 0 such that θ is continuous in Ci-topology on forms with support in K. REMARK: A smooth (p, q)-form ψ defines a (p, q)-current: given a test- form α ∈ Λn−p,n−q

c

(M), consider the functional α − →

• M ψ ∧ α. This gives an

embedding Λp,q(M) ֒ → Dp,q(M) from forms to currents. REMARK: Currents are (p, q)-forms with coefficients in generalized functions. 4

Hodge theory, lecture 16

• M. Verbitsky

Cohomology of currents DEFINITION: Define the de Rham differential on the space of currents using the formula dψ, α := −(−1) ˜

ψψ, dα. This definition is compatible

with the embedding Λp,q(M) ֒ → Dp,q(M) from forms to currents:

• M dψ ∧ α =
• M d(ψ ∧ α) − (−1) ˜

ψ

• M ψ ∧ dα = −(−1) ˜

ψ

• M ψ ∧ dα

by Stokes’ formula. REMARK: The Dolbeault differentials ∂ = d1,0, ∂ = d0,1 are defined on currents using the same formula. EXERCISE: Prove the Poincar´ e lemma for currents. DEFINITION: Let f : X − → Y be a proper holomorphic map of complex manifolds, dimC X = dimC Y + k, and α a (p, q)-current on X. Define the pushforward f∗α using f∗α, τ := α, f∗τ, where τ is any test-form. Then f∗α has bidimension (p − k, q − k). One should think of f∗ as of fiberwise integration. REMARK: Clearly, d f∗α = f∗dα, ∂f∗α = f∗∂α, and so on. REMARK: Pullback of currents is (generally speaking) not well-defined. 5

Hodge theory, lecture 16

• M. Verbitsky

Poincar´ e-Lelong formula CLAIM: (Poincar´ e-Lelong formula) Consider a current on C given by

1 πzdz. Then d

1

πzdz

• = δ0 Vol, where δ0 is

δ-function in 0. Proof: For any function smooth f on a closure of a disc D and w ∈ D, Cauchy formula gives f(w) = 1 2π√−1

• ∂D

f(z) z − wdz − 1 π

• D

∂f z − w ∧ dz. Applying this to a test-function f with compact support inside D, we obtain f(w) = −

1

πzdz, ∂f

• =
• ∂

1

πz

• dz, f
• =
• d

dz

πz

• , f
• .

(the last equality is true because dη = ∂η for any (1, 0)-form on a disc). 6

Hodge theory, lecture 16

• M. Verbitsky

Poincar´ e-Dolbeault-Grothendieck (dimension 1) COROLLARY: Let π1, π2 : C2 − → C be coordinate projections, and ξ a (1,0)-current on C2 defined by ξ :=

1 π(z−w)dw, where w, z are coordinates on

• C2. Consider convolution with the current ξ, given by Pξ(τ) := π2∗(π∗

1τ ∧ξ).

Then ∂Pξ(α) = α for any (0, 1)-form α with compact support. Proof: ∂Pξ(α) = π2∗(π∗

1α ∧ ∂ξ) = π2∗(π∗ 1α ∧ δ△) = α, where δ△ is δ-function

• f the diagonal △, which is defined as κ, δ△ :=
• △ κ.

COROLLARY: For any (0,1)-form α with compact support on C there exists a function f ∈ C∞(C) such that ∂f = α. Moreover, f can be chosen in such a way that |f(z)| < C 1

|z| for some constant C > 0 depending on

• C |α|.

Proof: Take f = Pξ(α). From the definition of Pξ we obtain |f(z)| <

dist(z, S)−1

C |α|, where S = Supp(α). This implies the estimate.

REMARK: Similarly, for any (1, 1)-form α with compact support one has ∂(Pξ(α)) = α, with the same asymptotic estimates on Pξ(α). 7